## Abstract

We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier–Stokes equations satisfying certain critical conditions. The solutions we consider have \(\Vert r^{1-\frac{3}{q}}u\Vert _{L_t^\infty L_x^q}<\infty \) where \(r=\sqrt{x_1^2+x_2^2}\) and either \(q\in (3,\infty )\), or *u* is axisymmetric and \(q\in (2,3]\). Using the strategy of Tao (Quantitative bounds for critically bounded solutions to the Navier–Stokes equations. arXiv:1908.04958, 2019), we obtain improved subcritical estimates for such solutions depending only on the double exponential of the critical norm. One consequence is a double logarithmic lower bound on the blowup rate. We make use of some tools such as a decomposition of the solution that allows us to use energy methods in these spaces, as well as a Carleman inequality for the heat equation suited for proving quantitative backward uniqueness in cylindrical regions.

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## Notes

We thank the anonymous referee for these references.

## References

Albritton, D.: Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces.

*Anal. PDE***11**(6), 1415–1456, 2018Albritton, D., Barker, T.: Global weak Besov solutions of the Navier-Stokes equations and applications.

*Arch. Ration. Mech. Anal.***232**(1), 197–263, 2019Barker, T., Prange, C.: Mild criticality breaking for the Navier–Stokes equations.

*J. Math. Fluid Mech.***23**(3), 1–12, 2021Barker, T., Prange, C.: Quantitative regularity for the Navier-Stokes equations via spatial concentration.

*Commun. Math. Phys.*1–76, 2021Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations.

*Commun. Math. Phys.***94**(1), 61–66, 1984Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations.

*Commun. Pure Appl. Math.***35**(6), 771–831, 1982Chae, D., Lee, J.: On the regularity of the axisymmetric solutions of the Navier–Stokes equations.

*Math. Z.***239**(4), 645–671, 2002Chemin, J.-Y., Planchon, F.: Self-improving bounds for the Navier–Stokes equations.

*Bull. Soc. Math. France***140**(4), 583–597, 2012Chen, C.-C., Strain, R.M., Tsai, T.-P., Yau, H.-T.: Lower bounds on the blow-up rate of the axisymmetric Navier–Stokes equations II.

*Commun. Partial Differ. Equ.***34**(3), 203–232, 2009Chen, H., Fang, D., Zhang, T.: Regularity of 3D axisymmetric Navier–Stokes equations.

*Discrete Contin. Dyn. Syst.***37**(4), 2017, 1923Escauriaza, L., Seregin, G.A., Šverák, V.: \(L_{3,\infty }\)-solutions of the Navier–Stokes equations and backward uniqueness.

*Russ. Math. Surv.***58**(2), 211–250, 2003Gallagher, I., Koch, G.S., Planchon, F.: A profile decomposition approach to the \(L_t^\infty (L_x^3)\) Navier–Stokes regularity criterion.

*Math. Ann.***355**(4), 1527–1559, 2013Gallagher, I., Koch, G.S., Planchon, F.: Blow-up of critical Besov norms at a potential Navier–Stokes singularity.

*Commun. Math. Phys.***343**(1), 39–82, 2016Koch, G., Nadirashvili, N., Seregin, G.A., Šverák, V., et al.: Liouville theorems for the Navier–Stokes equations and applications.

*Acta Math.***203**(1), 83–105, 2009Ladyzhenskaya, O.A.: On the uniqueness and on the smoothness of weak solutions of the Navier–Stokes equations.

*Zap. Nauchnykh Semin. POMI***5**, 169–185, 1967Lei, Z., Zhang, Q.S.: A Liouville theorem for the axially-symmetric Navier–Stokes equations.

*J. Funct. Anal.***261**(8), 2323–2345, 2011Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace.

*Acta Math.***63**, 193–248, 1934Maremonti, P., Shimizu, S.: Global existence of solutions to 2-D Navier–Stokes flow with non-decaying initial data in exterior domains.

*J. Math. Fluid Mech.***20**(3), 899–927, 2018Pan, X.: Regularity of solutions to axisymmetric Navier–Stokes equations with a slightly supercritical condition.

*J. Differ. Equ.***260**(12), 8485–8529, 2016Prodi, G.: Un teorema di unicita per le equazioni di Navier–Stokes.

*Ann. Mat. Appl.***48**(1), 173–182, 1959Seregin, G.: A certain necessary condition of potential blow up for Navier–Stokes equations.

*Commun. Math. Phys.***3**(312), 833–845, 2012Seregin, G.: Local regularity of axisymmetric solutions to the Navier–Stokes equations.

*Anal. Math. Phys.***10**(4), 1–20, 2020Seregin, G., Šverák, V.: On type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations.

*Commun. Partial Differ. Equ.***34**(2), 171–201, 2009Seregin, G.A., Zhou, D.: Regularity of solutions to the Navier–Stokes equations in \(B_{\infty ,\infty }^{-1}\).

*Zap. Nauchn. Sem. POMI***477**, 119–128, 2018Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Mathematics Division, Air Force Office of Scientific Research, 1961

Tao, T.: Localisation and compactness properties of the Navier–Stokes global regularity problem.

*Anal. PDE***6**(1), 25–107, 2013Tao, T.: Quantitative bounds for critically bounded solutions to the Navier–Stokes equations. arXiv preprint arXiv:1908.04958, 2019

## Acknowledgements

The author is grateful to Terence Tao for many helpful discussions, as well as for providing feedback on a version of the manuscript. We also thank the anonymous referee for their careful reading and comments. This work was partially funded by NSF Grant DMS-1764034.

## Funding

This work was partially funded by NSF Grant DMS-1764034.

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Palasek, S. Improved Quantitative Regularity for the Navier–Stokes Equations in a Scale of Critical Spaces.
*Arch Rational Mech Anal* **242**, 1479–1531 (2021). https://doi.org/10.1007/s00205-021-01709-5

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DOI: https://doi.org/10.1007/s00205-021-01709-5